guide:0f7df15c41

[math] \newcommand{\mathds}{\mathbb}[/math]

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

Welcome to advanced operator algebra theory, again. What we saw in the previous chapter was in fact just half of the story, and the other half, regarding hyperfiniteness, still remains to be told. The idea indeed is that there has been a considerable amount of work on hyperfiniteness, comparable in size and difficulty with the general classification work for the factors, based on reduction theory, and we will discuss this here.

In practice, all this will be quite independent from what we did in chapter 11. What we have to do is to go back to the functional analysis methods for general von Neumann algebras developed in chapter 9, and to the theory of factors, and notably of the type [math]{\rm II}_1[/math] factors developed in chapter 10, with the aim of further building on this. Following old, classical work of Murray-von Neumann , our main object of study will be the central example of a [math]{\rm II}_1[/math] factor, namely the “smallest” one, the hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math].

Once this factor [math]R[/math] introduced, and its basic theory understood, we will go on a more advanced discussion, including more theory of [math]R[/math], following Connes , then a discussion of various quantum group aspects, as a continuation of what has been said in chapter 10, and finally a discussion of the connections with the material in chapter 11.

Needless to say, this chapter will be a bit like the previous one, more of a survey. Also, let us mention that afterwards, in chapters 13-16 below, we will go back to a more normal pace, with a standard introduction to the Jones theory of inclusions of [math]{\rm II}_1[/math] factors, with full details. The notion of hyperfiniteness and the factor [math]R[/math] will of course show up there, every now and then, but usually at the end of each chapter, and most of the time using actually only its basic theory, and not most of the advanced material below.

In order to get started now, let us formulate the following definition:

Definition

A von Neumann algebra [math]A\subset B(H)[/math] is called hyperfinite when it appears as the weak closure of an increasing limit of finite dimensional algebras:

[[math]] A=\overline{\bigcup_iA_i}^{\,w} [[/math]]

When [math]A[/math] is a [math]{\rm II}_1[/math] factor, we call it hyperfinite [math]{\rm II}_1[/math] factor, and we denote it by [math]R[/math].

As a first observation, there are many hyperfinite von Neumann algebras, for instance because any finite dimensional von Neumann algebra [math]A=\oplus_iM_{n_i}(\mathbb C)[/math] is such an algebra, as one can see simply by taking [math]A_i=A[/math] for any [math]i[/math], in the above definition.

Also, given a measured space [math]X[/math], by using a dense sequence of points inside it, we can write [math]X=\bigcup_iX_i[/math] with [math]X_i\subset X[/math] being an increasing sequence of finite subspaces, and at the level of the corresponding algebras of functions this gives a decomposition as follows, which shows that the algebra [math]A=L^\infty(X)[/math] is hyperfinite, in the above sense:

[[math]] L^\infty(X)=\overline{\bigcup_iL^\infty(X_i)}^{\,w} [[/math]]

The interesting point, however, is that when trying to construct [math]{\rm II}_1[/math] factors which are hyperfinite, all the possible constructions lead in fact to the same factor, denoted [math]R[/math]. This is an old theorem of Murray and von Neumann , that we will explain now.

In order to get started, we will need a number of technical ingredients. Generally speaking, out main tool will be the expectation [math]E_i:A\to A_i[/math] from a hyperfinite von Neumann algebra [math]A[/math] onto its finite dimensional subalgebras [math]A_i\subset A[/math], so talking about such conditional expectations will be our first task. Let us start with:

Proposition

Given an inclusion of finite von Neumann algebras [math]A\subset B[/math], there is a unique linear map

[[math]] E:B\to A [[/math]]

which is positive, unital, trace-preserving and satisfies the following condition:

[[math]] E(b_1ab_2)=b_1E(a)b_2 [[/math]]

This map is called conditional expectation from [math]B[/math] onto [math]A[/math].

Show Proof

We make use of the standard representation of the finite von Neumann algebra [math]B[/math], with respect to its trace [math]tr:B\to\mathbb C[/math], as constructed in chapter 10:

[[math]] B\subset L^2(B) [[/math]]

If we denote by [math]\Omega[/math] the cyclic and separating vector of [math]L^2(B)[/math], we have an identification of vector spaces [math]A\Omega=L^2(A)[/math]. Consider now the following orthogonal projection:

[[math]] e:L^2(B)\to L^2(A) [[/math]]

It follows from definitions that we have an inclusion [math]e(B\Omega)\subset A\Omega[/math], and so our projection [math]e[/math] induces by restriction a certain linear map, as follows:

[[math]] E:B\to A [[/math]]

This linear map [math]E[/math] and the orthogonal projection [math]e[/math] are then related by:

[[math]] exe=E(x)e [[/math]]

But this shows that the linear map [math]E[/math] satisfies the various conditions in the statement, namely positivity, unitality, trace preservation and bimodule property. As for the uniqueness assertion, this follows by using the same argument, applied backwards, the idea being that a map [math]E[/math] as in the statement must come from the projection [math]e[/math].

■

Following Jones , who was a heavy user of such expectations, we will be often interested in what follows in the orthogonal projection [math]e:L^2(B)\to L^2(A)[/math] producing the expectation [math]E:B\to A[/math], rather than in [math]E[/math] itself. So, let us formulate:

Definition

Associated to any inclusion of finite von Neumann algebras [math]A\subset B[/math], as above, is the orthogonal projection

[[math]] e:L^2(B)\to L^2(A) [[/math]]

producing the conditional expectation [math]E:B\to A[/math] via the following formula:

[[math]] exe=E(x)e [[/math]]

This projection is called Jones projection for the inclusion [math]A\subset B[/math].

We will heavily use Jones projections in chapters 13-16 below, in the context where both the algebras [math]A,B[/math] are [math]{\rm II}_1[/math] factors, when systematically studying the inclusions of such [math]{\rm II}_1[/math] factors [math]A\subset B[/math], called subfactors. In connection with our present hyperfiniteness questions, the idea, already mentioned above, will be that of using the conditional expectation [math]E_i:A\to A_i[/math] from a hyperfinite von Neumann algebra [math]A[/math] onto its finite dimensional subalgebras [math]A_i\subset A[/math], as well as its Jones projection versions [math]e_i:L^2(A)\to L^2(A_i)[/math]. Let us start with a technical approximation result, as follows:

Proposition

Assume that a von Neumann algebra [math]A\subset B(H)[/math] appears as an increasing limit of von Neumann subalgebras

[[math]] A=\overline{\bigcup_iA_i}^{\,w} [[/math]]

and denote by [math]E_i:A\to A_i[/math] the corresponding conditional expectations.

• We have [math]||E_i(x)-x||\to0[/math], for any [math]x\in A[/math].
• If [math]x_i\in A_i[/math] is a bounded sequence, satisfying [math]x_i=E_i(x_{i+1})[/math] for any [math]i[/math], then this sequence has a norm limit [math]x\in A[/math], satisfying [math]x_i=E_i(x)[/math] for any [math]i[/math].

Show Proof

Both the assertions are elementary, as follows:

(1) In terms of the Jones projections [math]e_i:L^2(A)\to L^2(A_i)[/math] associated to the expectations [math]E_i:A\to A_i[/math], the fact that the algebra [math]A[/math] appears as the increasing union of its subalgebras [math]A_i[/math] translates into the fact that the [math]e_i[/math] are increasing, and converging to [math]1[/math]:

[[math]] e_i\nearrow1 [[/math]]

But this gives [math]||E_i(x)-x||\to0[/math], for any [math]x\in A[/math], as desired.

(2) Let [math]\{x_i\}\subset A[/math] be a sequence as in the statement. Since this sequence was assumed to be bounded, we can pick a weak limit [math]x\in A[/math] for it, and we have then, for any [math]i[/math]:

[[math]] E_i(x)=x_i [[/math]]

Now by (1) we obtain from this [math]||x-x_n||\to0[/math], which gives the result.

■

We have now all the needed ingredients for formulating a first key result, in connection with the hyperfinite [math]{\rm II}_1[/math] factors, due to Murray-von Neumann , as follows:

Proposition

Given an increasing union on matrix algebras, the following construction produces a hyperfinite [math]{\rm II}_1[/math] factor

[[math]] R=\overline{\bigcup_{n_i}M_{n_i}(\mathbb C)}^{\,w} [[/math]]

called Murray-von Neumann hyperfinite factor.

Show Proof

This basically follows from the above, in two steps, as follows:

(1) The von Neumann algebra [math]R[/math] constructed in the statement is hyperfinite by definition, with the remark here that the trace on it [math]tr:R\to\mathbb C[/math] comes as the increasing union of the traces on the matrix components [math]tr:M_{n_i}(\mathbb C)\to\mathbb C[/math], and with all the details here being elementary to check, by using the usual standard form technology.

(2) Thus, it remains to prove that [math]R[/math] is a factor. For this purpose, pick an element belonging to its center, [math]x\in Z(R)[/math], and consider its expectation on [math]A_i=M_{n_i}(\mathbb C)[/math]:

[[math]] x_i=E_i(x) [[/math]]

We have then [math]x_i\in Z(A_i)[/math], and since the matrix algebra [math]A_i=M_{n_i}(\mathbb C)[/math] is a factor, we deduce from this that this expected value [math]x_i\in A_i[/math] is given by:

[[math]] x_i=tr(x_i)1=tr(x)1 [[/math]]

On the other hand, Proposition 12.4 applies, and shows that we have:

[[math]] ||x_i-x||=||E_i(x)-x||\to0 [[/math]]

Thus our element is a scalar, [math]x=tr(x)1[/math], and so [math]R[/math] is a factor, as desired.

■

Next, we have the following substantial improvement of the above result, also due to Murray-von Neumann , which will be our final saying on the subject:

Theorem

There is a unique hyperfinite [math]{\rm II}_1[/math] factor, called Murray-von Neumann hyperfinite factor [math]R[/math], which appears as an increasing union on matrix algebras,

[[math]] R=\overline{\bigcup_{n_i}M_{n_i}(\mathbb C)}^{\,w} [[/math]]

with the isomorphism class of this union not depending on the exact sizes of the matrix algebras involved, nor on the particular inclusions between them.

Show Proof

We already know from Proposition 12.5 that the union in the statement is a hyperfinite [math]{\rm II}_1[/math] factor, for any choice of the matrix algebras involved, and of the inclusions between them. Thus, in order to prove the result, it all comes down in proving the uniqueness of the hyperfinite [math]{\rm II}_1[/math] factor. But this can be proved as follows:

(1) Given a [math]{\rm II}_1[/math] factor [math]A[/math], a von Neumann subalgebra [math]B\subset A[/math], and a subset [math]S\subset A[/math], let us write [math]S\subset_\varepsilon B[/math] when the following condition is satisfied, with [math]||x||_2=\sqrt{tr(x^*x)}[/math]:

[[math]] \forall x\in S,\exists y\in B, ||x-y||_2\leq\varepsilon [[/math]]

With this convention made, given a [math]{\rm II}_1[/math] factor [math]A[/math], the fact that this factor is hyperfinite in the sense of Definition 12.1 tells us that for any finite subset [math]S\subset A[/math], and any [math]\varepsilon \gt 0[/math], we can find a finite dimensional von Neumann subalgebra [math]B\subset A[/math] such that:

[[math]] S\subset_\varepsilon B [[/math]]

(2) With this observation made, assume that we are given a hyperfinite [math]{\rm II}_1[/math] factor [math]A[/math]. Let us pick a dense sequence [math]\{x_k\}\subset A[/math], and let us set:

[[math]] S_k=\{x_1,\ldots,x_k\} [[/math]]

By choosing [math]\varepsilon=1/k[/math] in the above, we can find, for any [math]k\in\mathbb N[/math], a finite dimensional von Neumann subalgebra [math]B_k\subset A[/math] such that the following condition is satisfied:

[[math]] S_k\subset_{1/k}B_k [[/math]]

(3) Our first claim is that, by suitably choosing our subalgebra [math]B_k\subset A[/math], we can always assume that this is a matrix algebra, of the following special type:

[[math]] B_k=M_{2^{n_k}}(\mathbb C) [[/math]]

But this is something which is quite routine, which can be proved by starting with a finite dimensional subalgebra [math]B_k\subset A[/math] as above, and then perturbing its set of minimal projections [math]\{e_i\}[/math] into a set of projections [math]\{e_i'\}[/math] which are close in norm, and have as traces multiples of [math]2^n[/math], with [math]n \gt \gt 0[/math]. Indeed, the algebra [math]B_k'\subset A[/math] having these new projections [math]\{e_i'\}[/math] as minimal projections will be then arbitrarily close to the algebra [math]B_k[/math], and so will still contain the subset [math]S_k[/math] in the above approximate sense, and due to our trace condition, will be contained in a subalgebra of type [math]B_k''\simeq M_{2^{n_k}}(\mathbb C)[/math], as desired.

(4) Our next claim, whose proof is similar, by using standard perturbation arguments for the corresponding sets of minimal projections, is that in the above the sequence of subalgebras [math]\{B_k\}[/math] can be chosen increasing. Thus, up to a rescaling of everything, we can assume that our sequence of subalgebras [math]\{B_k\}[/math] is as follows:

[[math]] B_k=M_{2^k}(\mathbb C) [[/math]]

(5) But this finishes the proof. Indeed, according to the above, we have managed to write our arbitrary hyperfinite [math]{\rm II}_1[/math] factor [math]A[/math] as a weak limit of the following type:

[[math]] A=\overline{\bigcup_kM_{2^k}(\mathbb C)}^{\,w} [[/math]]

Thus we have uniqueness indeed, and our result is proved.

■

The above result is something quite fundamental, and adds to a series of similar results, or rather philosophical conclusions, which are quite surprising, as follows:

(1) We have seen early on in this book that, up to isomorphism, there is only one Hilbert to be studied, namely the infinite dimensional separable Hilbert space, which can be taken to be, according to knowledge and taste, either [math]H=L^2(\mathbb R)[/math], or [math]H=l^2(\mathbb N)[/math].

(2) Regarding now the study of the operator algebras [math]A\subset B(H)[/math] over this unique Hilbert space, another somewhat surprising conclusion, from chapter 6, is that we won't miss much by assuming that [math]A=M_N(L^\infty(X))[/math] is a random matrix algebra.

(3) And now, guess what, what we just found is that when trying to get beyond random matrices, and what can be done with them, we are led to yet another unique von Neumann algebra, namely the above Murray-von Neumann hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math].

(4) And for things to be complete, we will see later that when getting beyond type [math]{\rm II}_1[/math], things won't change, because the other types of hyperfinite factors, not necessarily of type [math]{\rm II}_1[/math], can be all shown to ultimately come from [math]R[/math], via various constructions.

All this is certainly quite interesting, philosophically speaking. All in all, always the same conclusion, no need to go far to get to interesting algebras and questions: these interesting algebras and questions are just there, the most obvious ones.

Now back to more concrete things, one question is about how to best think of [math]R[/math], with Theorem 12.6 as stated not providing us with an answer. To be more precise, we would like to know what is the “best model” for [math]R[/math], that is, what exact matrix algebras should we use in practice, and with which inclusions between them. And here, a look at the proof of Theorem 12.6 suggests that the “best writing” of [math]R[/math] is as follows:

[[math]] R=\overline{\bigcup_kM_{2^k}(\mathbb C)}^{\,w} [[/math]]

And we can in fact do even better, by observing that the inclusions between matrix algebras of size [math]2^k[/math] appear via tensor products, and formulating things as follows:

Proposition

The hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math] appears as

[[math]] R=\overline{\bigotimes_{r\in\mathbb N}M_2(\mathbb C)}^{\,w} [[/math]]

with the infinite tensor product being defined as an inductive limit, in the obvious way.

Show Proof

This follows from the above discussion, and with the remark that there is a binary choice there, of left/right type, to be made when constructing the inductive limit. And we prefer here not to make any choice, and leave things like this, because the best choice here always depends on the precise applications that you have in mind.

■

Along the same lines, we can ask as well for precise group algebra models for the hyperfinite [math]{\rm II}_1[/math] factor, [math]R=L(\Gamma)[/math], and the canonical choice here is as follows:

Proposition

The hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math] appears as

[[math]] R=L(S_\infty) [[/math]]

with [math]S_\infty=\bigcup_{r\in\mathbb N}S_r[/math] being the infinite symmetric group.

Show Proof

Consider indeed the infinite symmetric group [math]S_\infty[/math], which is by definition the group of permutations of [math]\{1,2,3,\ldots\}[/math] having finite support. Since such an infinite permutation with finite support must appear by extending a certain finite permutation [math]\sigma\in S_r[/math], with fixed points outside [math]\{1,\ldots,r\}[/math], we have then, as stated:

[[math]] S_\infty=\bigcup_{r\in\mathbb N}S_r [[/math]]

But this shows that the von Neumann algebra [math]L(S_\infty)[/math] is hyperfinite. On the other hand [math]S_\infty[/math] has the ICC property, and so [math]L(S_\infty)[/math] is a [math]{\rm II}_1[/math] factor. Thus, [math]L(S_\infty)=R[/math].

■

There are of course some more things that can be said here, because other groups of the same type as [math]S_\infty[/math], namely appearing as increasing limits of finite subgroups, and having the ICC property, will produce as well the hyperfinite factor, [math]L(\Gamma)=R[/math], and so there is some group theory to be done here, in order to fully understand such groups. However, we prefer to defer the discussion for later, after learning about amenability, which will lead to a substantial update of our theory, making such things obsolete.

As an interesting consequence of all this, however, let us formulate:

Proposition

Given two groups [math]\Gamma,\Gamma'[/math], each having the ICC property, and each appearing as an increasing union of finite subgroups, we have

[[math]] L(\Gamma)\simeq L(\Gamma') [[/math]]

while the corresponding group algebras might not be isomorphic, [math]\mathbb C[\Gamma]\neq\mathbb C[\Gamma'][/math].

Show Proof

Here the first assertion follows from the above discusssion, the von Neumann algebra in question being the hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math]. As for the last assertion, there are countless counterexamples here, all coming from basic group theory.

■

The point with the above result is that the isomorphisms of type [math]L(\Gamma)\simeq L(\Gamma')[/math] are in general impossible to prove with bare hands. Thus, we can see here the power of the Murray-von Neumann results in . And we can also see the magic of the weak topology, which by some kind of miracle, makes everyone equal in the end.

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].